In the preface, the author states that this text grew out of two upper-level courses she taught, one in linear algebra and the other in combinatorial matrix theory. She taught both courses from her own notes because she could not find a text to her liking. The author used a selection of topics from those notes for a new advanced linear algebra text. Quoting from the preface, “My hope is that it will be useful to those (students and professionals) who have taken a first linear algebra course but need to know more.”

The first three chapters are a quick review of introductory linear algebra, that is, vectors and matrices, inner products, and eigenvalues, followed by seven chapters on more advanced topics, similarity and canonical forms, Hermitian matrices, matrix and vector norms, and field of values. These chapters are well written with many examples. There is a nice excursion into Hilbert spaces and Fourier series, both finite and infinite dimensional. There are about a dozen exercises at the end of each chapter, and even more for the first three chapters. Some readers might skip to the exercises in the early chapters and test their grasp of basic linear algebra.

The next eight chapters present more specialized topics: the theorems of McCoy and Motzkin-Taussky on simultaneous triangularization; combinatorial results on block designs and 0-1 matrices, including Hadamard matrices; and non-negative matrices and Perron-Frobenius theory. Finally, the book ends with chapters on error-correcting codes and linear dynamical systems.

These later chapters continue in the style of the earlier parts, with some exceptions. In the chapters on non-negative matrices, Wielandt’s approach to irreducibility using directed graphs moves smoothly into primitive matrices and the Perron theorem. The chapter on block designs, however, should show more clearly the strategy for the proof of the Bruck-Ryser-Chowla theorem. The first part of the theorem could be proved early in the chapter. This would provide motivation for the later material that is necessary to complete the proof.

The preface asks: “Do we need another linear algebra book?” The answer is yes, of course, because the appropriate topics for a first or second undergraduate linear algebra course are always changing. Determinants have almost disappeared while the singular value decomposition only started to appear in the 1960s. This book does introduce topics in matrix theory, which may not normally appear in a standard second course in linear algebra, for example, Witt’s cancellation theorem. At the same time, it includes other topics that naturally involve matrices, such as graphs and block designs. I think this text would be useful in an upper-level matrix theory course, either as the primary text or as a source for topics in a new course.